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Hence, it is technically more correct to discuss singular points of a smooth mapping here rather than a singular point of a curve. The above definitions can be extended to cover implicit curves which are defined as the zero set of a smooth function, and it is not necessary just to consider algebraic varieties. The definitions can be ...
The pinch point (in this case the origin) is a limit of normal crossings singular points (the -axis in this case). These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow-up the whole v {\displaystyle v} -axis and not only the pinch point.
The Whitney umbrella x 2 = y 2 z has singular set the z axis, most of whose point are ordinary double points, but there is a more complicated pinch point singularity at the origin, so blowing up the worst singular points suggests that one should start by blowing up the origin. However blowing up the origin reproduces the same singularity on one ...
Point a is an ordinary point when functions p 1 (x) and p 0 (x) are analytic at x = a. Point a is a regular singular point if p 1 (x) has a pole up to order 1 at x = a and p 0 has a pole of order up to 2 at x = a. Otherwise point a is an irregular singular point.
It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves. It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
The point is a pole or non-essential singularity of if there exists a holomorphic function defined on with () nonzero, and a natural number such that () = () for all in {}. The least such number n {\displaystyle n} is called the order of the pole .
Antipodal point, the point diametrically opposite to another point on a sphere, such that a line drawn between them passes through the centre of the sphere and forms a true diameter; Conjugate point, any point that can almost be joined to another by a 1-parameter family of geodesics (e.g., the antipodes of a sphere, which are linkable by any ...
An algebraic variety that has no singular point is said to be non-singular or smooth. The concept is generalized to smooth schemes in the modern language of scheme theory . The plane algebraic curve (a cubic curve ) of equation y 2 − x 2 ( x + 1) = 0 crosses itself at the origin (0, 0) .